Department of Mathematics and Statistics

Friday, November 24th, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Jeffery Rosenthal, University of Toronto

Title: Adaptive MCMC for Everyone

Abstract: Markov chain Monte Carlo (MCMC) algorithms, such as the Metropolis Algorithm and the Gibbs Sampler, are an extremely useful and popular method of approximately sampling from complicated probability distributions. Adaptive MCMC attempts to automatically modify the algorithm while it runs, to improve its performance on the fly. However, such adaptation often destroys the ergodicity properties necessary for the algorithm to be valid. In this talk, we first illustrate MCMC algorithms using simple graphical Java applets. We then discuss adaptive MCMC, and present examples and theorems concerning its ergodicity and efficiency. We close with some recent ideas which make adaptive MCMC more widely applicable in broader contexts.

Wednesday, November 22nd, 2017

Time: 1:30 p.m.  Place: Jeffery Hall 319

Speaker: Jung-Jo Lee

Title: Cyclotomic units, values of the Riemann zeta function and p-adic measures

Abstract:  I would explain the connection between the cyclotomic units and the values of complex Riemann zeta function via higher logarithmic derivative maps. We can use Mahler transform to construct a p-adic zeta function.

Note: I consider a series of talks, each "hopefully" self-contained.

Talk 2 : Euler system of cyclotomic units and Iwasawa's main conjecture

Talk 3 : Euler system of Heegner points and Birch and Swinnerton-Dyer conjecture

Friday, November 17th, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Mokshay Madiman

Title: The convexifying effect of Minkowski summation

Abstract: For a compact subset $A$ of $R^d$, let $A(k)$ be the Minkowski sum of k copies of A, scaled by $1/k$. By a 1969 theorem of Emerson, Folkmann, Greenleaf, Shapley and Starr, $A(k)$ approaches the convex hull of A in Hausdorff distance as k goes to infinity; this fact has important applications in a number of areas including mathematical economics. A few years ago, the speaker conjectured that the volume of A(k) is non-decreasing in k, or in other words, that when the volume deficit between the convex hull of A and A(k) goes to 0, it actually does so monotonically. While this conjecture holds true in dimension 1 (as independently observed by F. Barthe), we show that it fails in dimension 12 or greater. Then we consider whether one can have monotonicity of convergence of when non-convexity is measured in alternate ways. Our main positive result is that Schneider’s index of non-convexity of A(k) converges monotonically to 0 as k increases; even the convergence does not seem to have been known before. As a by-product, we also obtain optimal rates of convergence. We also obtain analogous results for the Hausdorff distance to the convex hull, as well as for the inner radius, and demonstrate applications to discrepancy theory. Joint work with Matthieu Fradelizi (Marne-la-Vallée), Arnaud Marsiglietti (CalTech), and Artem Zvavitch (Kent State).

Mokshay Madiman (University of Delaware): Mokshay Madiman has been an Associate Professor in the Department of Mathematical Sciences at the University of Delaware since January 2013. Dr. Madiman received his Ph.D. degree in applied mathematics from Brown University in 2005. From 2005 to 2012, he worked at the Department of Statistics at Yale University, New Haven, CT, rst as a Gibbs Assistant Professor, then as an Assistant Professor, and nally as an Associate Professor of Statistics and Applied Mathematics. From 2014 to 2017, he was also an Adjunct Professor of Mathematics at the Tata Institute of Fundamental Research, Mumbai. He has spent a semester each in visiting positions at the Tata Institute; the Indian Institute of Science, Bangalore; Princeton University; and the Institute for Mathematics and its Applications, Minneapolis. Dr. Madiman's research is primarily in probability and information theory, but also interacts with combinatorics, functional analysis, and statistics.

Friday, November 10th, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Hector Pasten

Title: In Quest of Arithmetic Derivatives

Abstract: There are well-known arithmetic analogies between integers and polynomials (or more generally, holomorphic functions), specially with respect to solutions of Diophantine equations. A crucial aspect that is missing in this analogy is that for polynomials and holomorphic functions one can use derivatives, while for integers there is no direct substitute of this operation. After some motivation, I will discuss a couple of possible candidates for arithmetic derivatives.

Hector Pasten (Harvard University): Hector Pasten (Chile, 1988) is since 2014 a Benjamin Peirce Fellow at Harvard University. He was also a member of the Institute for Advanced Study at Princeton (2015-2016). Pasten received a Ph.D. in 2010 from Universidad de Concepcion in the subject of mathematical logic and non-archimedean analysis. Then he received a Ph.D. in 2014 from Queen's University in the subject of number theory. Among other distinctions, Pasten was awarded the Governor-General of Canada Academic Gold Medal (2014), the Doctoral Prize of the Canadian Mathematical Society (2015), and the Mathematical Council of the Americas Prize (2017). His current research is in number theory and related areas, addressing topics such as decidability of arithmetic structures, Diophantine approximation, and analogies between number theory and value distribution.

Friday, November 3rd, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Nasser Sadeghkhani

Title: Bayesian Predictive Density Estimation with Additional Information

Abstract: In the context of Bayesian theory and decision theory, the estimation of a predictive density of a random variable represents an important and challenging problem. Often the times there is some additional information at our disposal which is unduly being ignored. In this talk, we deal with strategies to take into account this kind of information, in order to obtain e ective and sometimes better performing predictive densities than others in the literature.

Nasser Sadeghkhani (Queen's University): Nasser Sadeghkhani earned his Ph.D in Mathematics (Statistics) from the Universite de Sherbrooke in 2017 under the supervision of Eric Marchand. He recently joined the Department of Mathematics and Statistics at Queen's University as a Coleman Postdoctoral Fellow. Dr Sadeghkhani's research interests include Bayesian Statistics, Multivariate Statistics, Survival Analysis, Decision Theory, and Predictive Inference.